Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28353 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,39)(36,40);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,27)(26,30)(28,36)(29,34)(31,38)(32,40)(33,35)(37,39);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,31)(30,38)(32,39)(37,40);(10,14)(15,21)(16,22)(23,24)(27,34)(30,38)(33,36)(37,40)Polycyclic presentation:
2^9|b^2=f,c^2=fh,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6861285);